Rebound Attack Florian Mendel Institute for Applied Information Processing and Communications (IAIK) Graz University of Technology Inffeldgasse 16a, A-8010 Graz, Austria http://www.iaik.tugraz.at/
Outline Motivation 1 2 Whirlpool Hash Function Application of the Rebound Attack 3 Summary 4
The Rebound Attack [MRST09] Tool in the differential cryptanalysis of hash functions Invented during the design of Grøstl AES-based designs allow a simple application of the idea Has been applied to a wide range of hash functions Echo, Grøstl, JH, Lane, Luffa, Maelstrom, Skein, Twister, Whirlpool, . . .
The Rebound Attack E bw E in E fw inbound outbound outbound Applies to block cipher and permutation based designs: E = E fw ◦ E in ◦ E bw P = P fw ◦ P in ◦ P bw
The Rebound Attack E bw E in E fw inbound outbound outbound Inbound phase efficient meet-in-the-middle phase in E in using available degrees of freedom Outbound phase probabilistic part in E bw and E fw repeat inbound phase if needed
The Whirlpool Hash Function M 1 M 2 M 3 M t f f f f H ( m ) IV designed by Barretto and Rijmen in 2000 [BR00] evaluated by NESSIE standardized by ISO/IEC 10118-3:2003 iterative, based on the Merkle-Damg˚ ard design principle message block, chaining values, hash size: 512 bit
The Whirlpool Compression Function key schedule H j − 1 SB SC MR AC state update M j H j SB SC MR AK 512-bit hash value and using 512-bit message blocks Block-cipher based design (similar to AES) Miyaguchi-Preneel mode with conservative key schedule
The Whirlpool Round Transformations SubBytes ShiftColumns MixRows AddRoundKey K i S(x) + The state update and the key schedule update an 8 × 8 state S and K of 64 bytes 10 rounds each AES like round transformation r i = AK ◦ MR ◦ SC ◦ SB
Notations Round i C i K SB K SC K MR K i − 1 K i i i i SB SC MR AC S SB S SC S MR S i − 1 S i i i i SB SC MR AK
Collision Attack on Whirlpool key schedule H j − 1 SB SC MR AC state update ∆ M j M j H j SB SC MR AK 1-block collision: fixed H j − 1 (to IV ) f ( M j , H j − 1 ) = f ( M ∗ j , H j − 1 ) , M j � = M ∗ j generic complexity 2 256 ( n = 512)
Collision Attack on 4 Rounds K 0 K 1 K 2 K 3 K 4 constant SB SB SB SB IV SC SC SC SC MR MR MR MR AC AC AC AC S 0 S 3 S 1 S 2 S 4 SB SB SB SB M 1 SC SC SC SC H 1 MR MR MR MR AK AK AK AK Differential trail with minimum number of active S-boxes 81 for any 4-round trail (1 → 8 → 64 → 8) maximum differential probability: ( 2 − 5 ) 81 = 2 − 405 How to find a message pair following the differential trail?
First: Use Truncated Differences S 0 S 1 S 2 S 3 S 4 SB SB SB SB M 1 SC SC SC SC H 1 MR MR MR MR AK AK AK AK byte-wise truncated differences: active / not active we do not mind about actual differences single active byte at input and output is enough probabilistic in MixRows: 2 − 56 for 8 → 1 we can remove many restrictions (more freedom) hopefully less complexity of message search
How to Find a Message Pair? S 0 S 1 S 2 S 3 S 4 SB SB SB SB M 1 SC SC SC SC H 1 MR MR MR MR AK AK AK AK message modification? inside out? meet in the middle? rebound!
Rebound Attack on 4 Rounds [MRST09] S 0 S 1 S 2 S 3 S 4 SB SB SB SB M 1 SC SC SC SC H 1 MR MR MR MR AK AK AK AK outbound phase inbound phase outbound phase Inbound phase (1) start with differences in round 2 and 3 (2) match-in-the-middle at S-box using values of the state Outbound phase (3) probabilistic propagation in MixRows in round 1 and 4 (4) match one-byte difference of feed-forward
Inbound Phase S SB S SC S MR S 2 2 3 3 ee ee ee 9f ee 23 71 c1 cd e8 f4 90 d4 75 1b 5e cd 3a 85 50 cc 6d 9a 49 43 c5 c0 0d cc 01 0a 70 43 e9 27 e6 ? MR a2 b1 63 11 96 1e 4d 04 SC b9 SB b1 60 20 f4 1e cd bf 10 5a AK MR f8 ed 85 b7 43 5a d5 fc 8c 16 27 51 43 15 de 2b f8 08 4d 34 96 90 f1 f8 07 5e c0 differences match differences differences get values get values (1) Start with arbitrary differences in state S MR 3 linearly propagate all differences backward to S SB 3 linearly propagate row-wise forward from S SC to S 2 2 (2) Match-in-the-middle at SubBytes layer check if differences can be connected (for each S-box) with probability 2 − xx we get 2 xx solutions for each row
Match-in-the-Middle for Single S-box ∆ a Sbox ∆ b Check for matching input/output differences Using Difference Distribution Table (DDT) Sbox ( x ) ⊕ Sbox ( x ⊕ ∆ a ) = ∆ b
Difference Distribution Table (Whirlpool) in \ out 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d 1e 1f 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 6 2 0 0 6 2 0 0 0 4 0 0 0 0 0 02 0 0 0 0 0 0 0 2 0 0 0 0 4 0 0 0 03 0 2 2 0 2 2 0 0 2 0 0 0 0 0 0 2 04 0 0 2 2 0 4 0 0 0 2 2 2 2 0 2 0 05 0 0 0 0 0 2 0 2 0 0 0 0 0 0 4 2 06 . 4 0 2 0 0 2 0 0 2 6 2 4 0 2 2 0 . 07 . 0 2 2 0 0 2 0 0 4 0 2 0 2 0 2 0 . 08 . 0 0 0 0 2 2 2 0 0 0 0 2 2 4 4 0 . 09 8 0 0 0 2 4 2 2 0 0 0 0 0 2 0 2 0a 0 0 0 0 2 0 2 0 2 0 2 0 0 0 0 0 0b 8 2 2 2 2 0 0 0 0 2 2 2 2 2 0 4 0c 0 2 2 0 0 0 0 4 0 2 2 0 0 2 4 2 0d 0 2 2 0 0 2 4 4 0 0 2 2 0 0 0 2 0e 4 0 4 2 0 0 0 0 2 0 2 0 4 2 0 0 0f 0 2 0 0 0 2 0 0 0 0 0 0 2 0 2 2 . . . Differences can be connected if there is a non-zero entry in the table
Match-in-the-Middle for Single S-box ∆ a Sbox ∆ b Check for matching input/output differences Using Difference Distribution Table (DDT) Sbox ( x ) ⊕ Sbox ( x ⊕ ∆ a ) = ∆ b Solve equation for all x and count the number of solutions
Difference Distribution Table (Whirlpool) The number of differentials and possible pairs for the Sbox solutions frequency 0 39655 2 20018 4 5043 6 740 8 79 256 1 25880/65025 entries (with ∆ a , ∆ b � = 0 ) in DDT are nonzero we get either 2, 4, 6 or 8 values for each match 25880 65025 · 65280 25880 = 1 . 004 values (right pairs) on average
Match-in-the-Middle for Single S-box ∆ a Sbox ∆ b Check for matching input/output differences Using Difference Distribution Table (DDT) Sbox ( x ) ⊕ Sbox ( x ⊕ ∆ a ) = ∆ b Solve equation for all x and count the number of solutions. ∼ 1 values (right pairs) on average
Inbound Phase S SB S SC S MR S 2 2 3 3 ee ee ee 9f ee 23 71 c1 cd e8 f4 90 d4 75 1b 5e cd 3a 85 50 cc 6d 9a 49 43 c5 c0 0d cc 01 0a 70 43 e9 27 e6 ? MR a2 b1 63 11 96 1e 4d 04 SC b9 SB b1 60 20 f4 1e cd bf 10 5a AK MR f8 ed 85 b7 43 5a d5 fc 8c 16 27 51 43 15 de 2b f8 08 4d 34 96 90 f1 f8 07 5e c0 differences match differences differences get values get values (1) Start with arbitrary differences in state S MR 3 linearly propagate all differences backward to S SB 3 linearly propagate row-wise forward from S SC to S 2 2 (2) Match-in-the-middle at SubBytes layer check if differences can be connected (for each S-box) we need to solve each row at once: complexity ∼ 2 10 . 6 (average 1)
Inbound Phase S SB S SC S MR S 2 2 3 3 ee ee ee 9f ee 23 71 c1 cd e8 f4 90 d4 75 1b 5e cd 3a 85 50 cc 6d 9a 49 43 c5 c0 0d cc 01 0a 70 43 e9 27 e6 MR a2 b1 63 11 96 1e 4d 04 SC b9 SB b1 60 20 f4 1e cd bf 10 5a AK MR f8 ed 85 b7 43 5a d5 fc 8c 16 27 51 43 15 de 2b f8 08 4d 34 96 90 f1 f8 07 5e c0 differences match differences differences get values get values (1) Start with arbitrary differences in state S MR 3 linearly propagate all differences backward to S SB 3 linearly propagate row-wise forward from S SC to S 2 2 (2) Match-in-the-middle at SubBytes layer check if differences can be connected (for each S-box) we need to solve each row at once: complexity ∼ 2 10 . 6 (average 1)
Outbound Phase S 0 S 1 S 2 S 3 S 4 SB SB SB SB SC SC SC SC MR MR MR MR AK AK AK AK outbound inbound outbound 2 − 56 average 1 2 − 56 (3) Propagate through MixRows of round 1 and round 4 using truncated differences (active bytes: 8 → 1) probability: 2 − 56 in each direction (4) Match difference in one active byte of feed-forward (2 − 8 ) ⇒ collision for 4 rounds of Whirlpool with complexity 2 120
Extending the Attack to 5 Rounds [LMR + 09] S 0 S 1 S 2 S 3 S 4 S 5 SB SB SB SB SB SC SC SC SC SC MR MR MR MR MR AK AK AK AK AK outbound inbound outbound 2 − 56 2 − 56 average 1 By adding one round in the inbound phase of the attack we can extend the attack to 5 rounds The outbound phase is identical to the attack on 4 rounds probability: 2 − 120 ⇒ Construct 2 120 starting points in the inbound phase with average complexity 1 (but increased memory of 2 64 )
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